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Journal of Physics A Mathematical and General

Spectral analysis ofq-oscillator with general bilinear interaction

Igor M. Loutsenko, Vyacheslav Pavlovich Spiridonov, Luc Vinet, Alexei S. Zhedanov

Published Nov 13, 1998• 6 citations
Publisher

Abstract

Spectra of the most general Hermitian Hamiltonian which is bilinear in the creation and annihilation operators of a q-harmonic oscillator are investigated with the help of the factorization method. It is shown that there are two factorization schemes leading to discrete spectra of complicated forms. For q-oscillator models with continuous spectrum the Hamiltonian may have normalizable eigenstates of infinite multiplicity. Existence of the continuous spectrum in the interacting system is also discussed.

DOI
10.1088/0305-4470/31/45/008

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Spectra of the most general Hermitian Hamiltonian which is bilinear in the creation and annihilation operators of a q-harmonic oscillator are investigated with the help of the factorization method. It is shown that there are two factorization schemes leading to discrete spectra of complicated forms. For q-oscillator models with continuous spectrum the Hamiltonian may have normalizable eigenstates of infinite multiplicity. Existence of the continuous spectrum in the interacting system is also discussed.
We investigate some aspects of q Heisenberg algebra. We show how su(2) and su(1,1) generators can be constructed in terms of the q creation and annihilation operators. We also construct the coherent states for the q oscillator and show that they can be obtained by the action of a displacement operator on the vacuum. For the multimode case with q=0, corresponding to the infinite statistics of Greenberg [Phys. Rev. Lett. 64, 705 (1990)], we generalize our single-mode construction to obtain the corresponding coherent states. These states, which are eigenstates of the annihilation operators, interestingly, turn out to be degenerate owing to the noncommutativity of the displacement operators.
Abstract Non-self-adjoint operators with a real discrete spectrum exhibit, in general, a quite different behavior from the one of self-adjoint operators, with the same spectrum. This fact has serious mathematical and physical implications, namely in the rigorous formulation of quantum mechanics (QM), where self-adjointness plays a central role, in particular in the spectral analysis of the involved operators. In this framework, a non-self-adjoint 2D-harmonic oscillator is considered. Its eigenvalues and eigenfunctions, as well as those of its adjoint, are explicitly determined. Although the eigensystems are complete they may not form bases, a drawback for the mathematical formulation of the QM of the system. The existence of a metric operator defining an inner product which renders the Hamiltonian self-adjoint, is discussed.
The author gives formulae for point spectra of Hermitian operators built from creation and annihilation type operators satisfying general algebraic relation which he calls (g, f, F) equivalence. By choosing the parameters in a specific way the description of the quantum q-oscillator is given in a model built from the q-derivative and dilatation operator. He shows the explicit form of eigenfunctions of the model from L 2 (R). Then he constructs the transformation between his model and the Q-oscillator with arbitrary Q real.
We propose an unified view to three well known quadratic Hamiltonians widely considered in quantum mechanics, i.e. those of the harmonic oscillator, of the inverted harmonic oscillator, and the Berry-Keating Hamiltonian. We describe an unified approach to determine the discrete spectrum of these operators, and to deduce their (generalized) eigenstates. We also look at these Hamiltonians as the generators of su(1,1)\mathfrak{su}(1,1)su(1,1), and of a generalized version of it relevant in presence of non Hermitian generators.
A system of harmonic oscillators in the presence of interaction, and with an arbitrary number of degrees of freedom, is considered. The most general form of the Hamiltonian is derived under the restriction that the states which are initially coherent remain coherent at all times. The equation of motion for the annihilation operator, obtained by using this Hamiltonian, is solved, and the frequency spectrum of the annihilation operator is discussed. By giving specific examples it is shown that, in general, the annihilation operators (or their eigenvalues) contain positive as well as negative frequency components and hence are not analytic signals. Some special cases are also considered where the annihilation operators are analytic signals.